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\begin{document}
\title{Math 301: Homework 2}
\author{Mary Radcliffe}
\date{due 16 Sept 2015}
\maketitle

Complete the following problems. Fully justify each response.

\begin{enumerate}

\item For $a, b\in \Z$, let $L(a, b)$ denote the set of lattice paths from $(0,0)$ to $(a, b)$.

Fix $n\in \Z$, and let $1\leq k\leq n-1$. Construct an injective function  \[F:L(k-1, n-k+1)\times L(k+1, n-k-1)\to L(k, n-k)\times L(k, n-k).\] Use this to prove that for all $k\leq n-1$, \[{n\choose k-1}{n\choose k+1}\leq {n\choose k}^2.\] (This property is called {\it log-concavity}. That is, a sequence $a_0, a_1, \dots, a_n$ is called log-concave if for all $1\leq k\leq n-1$, we have $a_{k-1}a_{k+1}\leq a_ka_k$. Hence, here we are proving that the sequence ${n\choose 0}, {n\choose 1}, \dots, {n\choose n}$ is log-concave.)

\item Prove that $\displaystyle\sum_{a_1+a_2+a_3=n} {n\choose a_1, a_2, a_3}(-1)^a_2 = 1$.

\item Suppose a class contains $2n$ students, and they have projects to work on in pairs.
\begin{enumerate}
\item How many different ways can the professor pair the students up?
\item After the first project, the professor wishes to create new pairings, so that nobody will be working with the same partner a second time. How many ways can the professor create this second set of pairings?
\end{enumerate}

\end{enumerate}

\end{document}